Preheat oven to 400 degrees F (200 degrees C). Grease the bottoms and sides of a 12-cup muffin tin.
Cream 1/3 cup butter, brown sugar, and cinnamon together. Drop 1 teaspoon into each muffin cup. Reserve the remaining mixture.
In a large bowl, combine flour, sugar, baking powder, and salt. Cut in 1/4 cup cold butter until crumbly. Make a well in center, and pour in the milk. Stir to form a soft dough, adding a bit more milk if needed.
Turn the dough out on lightly floured surface. Knead 8 to 10 times. Pat dough into rectangle about 1/3 inch thick and 12 inches long. Spread the remaining cinnamon mixture over dough rectangle. Sprinkle currants over top. Roll up like a jelly roll into a long log. Mark the dough first, to make sure you know where to cut, then cut the log into 12 slices. Place cut-side down in the prepared muffin pan.
Bake in in the preheated oven until golden brown, 20 to 25 minutes. Turn rolls out on a serving tray.
For the glaze, add milk to the confectioners' sugar and stir well, adding more liquid if necessary to make a thin glaze. Drizzle over cinnamon rolls.